Computing theta functions in quasi-linear time in genus 2 and above

Hugo Labrande 1, 2 Emmanuel Thomé 1
1 CARAMBA - Cryptology, arithmetic : algebraic methods for better algorithms
Inria Nancy - Grand Est, LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry
Abstract : We outline an algorithm to compute θ(z, τ) in genus 2 in quasi-optimal time, borrowing ideas from the algorithm for theta constants and the one for θ(z, τ) in genus 1. Our implementation shows a large speedup for precisions as low as a few thousand decimal digits. We also lay out a strategy to generalize this algorithm to genus g.
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Hugo Labrande, Emmanuel Thomé. Computing theta functions in quasi-linear time in genus 2 and above. LMS Journal of Computation and Mathematics, London Mathematical Society, 2016, Special issue: Algorithmic Number Theory Symposium XII, 19 (A), pp.163-177. ⟨10.1112/S1461157016000309⟩. ⟨hal-01277169⟩

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